The theory of filtrations of subalgebras, standardness and independence Anatoly M. Vershik∗ 24.01.2017 In memory of my friend Kolya K. (1933{2014) \Independence is the best quality, the best word in all languages." J. Brodsky. From a personal letter. Abstract The survey is devoted to the combinatorial and metric theory of fil- trations, i. e., decreasing sequences of σ-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of standardness, plays the role of a generalization of the no- tion of the independence of a sequence of random variables. We discuss the possibility of obtaining a classification of filtrations, their invariants, and various links to problems in algebra, dynamics, and combinatorics. Bibliography: 101 titles. Contents 1 Introduction 3 1.1 A simple example and a difficult question . .3 1.2 Three languages of measure theory . .5 1.3 Where do filtrations appear? . .6 1.4 Finite and infinite in classification problems; standardness as a generalization of independence . .9 arXiv:1705.06619v1 [math.DS] 18 May 2017 1.5 A summary of the paper . 12 2 The definition of filtrations in measure spaces 13 2.1 Measure theory: a Lebesgue space, basic facts . 13 2.2 Measurable partitions, filtrations . 14 2.3 Classification of measurable partitions . 16 ∗St. Petersburg Department of Steklov Mathematical Institute; St. Petersburg State Uni- versity Institute for Information Transmission Problems. E-mail: [email protected]. Par- tially supported by the Russian Science Foundation (grant No. 14-11-00581). 1 2.4 Classification of finite filtrations . 17 2.5 Filtrations we consider and how one can define them . 20 2.6 The equivalence relation associated with a filtration, cocycles and associated measures . 23 3 Tail filtrations in graded graphs and Markov chains 25 3.1 Path spaces of graded graphs (Bratteli diagrams) and Markov compacta . 25 3.2 Trees of paths . 26 3.3 An equipment of a multigraph and cotransition probabilities . 27 3.4 The Markov compactum of paths . 28 3.5 A cocycle of an equipped graph . 30 3.6 Examples of graded graphs and Markov compacta . 31 4 The Markov realization of locally finite filtrations 33 4.1 The theorem on a Markov realization . 33 4.2 Remarks on Markov models . 35 5 Standardness, criteria, finite isomorphism 37 5.1 The minimal model of a filtration . 37 5.2 Minimality of graphs, standardness, and the general standardness criterion . 40 5.3 The combinatorial standardness criterion . 42 5.4 The standardness criterion for an arbitrary filtration . 45 5.5 The martingale form of the standardness criterion and interme- diate conditions . 47 5.6 The spaces of filtrations, Markov compacta, and graded multigraphs 49 6 Nonstandardness: examples and a sketch of a general theory 51 6.1 Graphs of random walks over orbits of groups, graphs of words . 52 6.2 Walks over orbits of a free group . 54 6.3 Walks over orbits of free Abelian groups . 57 6.4 Lacunarity . 58 6.5 Scaled and secondary entropy . 59 6.6 A project of classification of nonstandard filtrations . 60 7 Projective limits of simplices, invariant measures, and the ab- solute of a graph 63 7.1 Projective limits of simplices, extreme points . 63 7.2 The fundamental problem: description of the absolute . 68 7.3 Several examples. Relation to the theory of locally finite filtrations 69 8 Historical commentary 71 9 Acknowledgements 74 2 1 Introduction This survey deals with an area in which measure theory is intertwined with combinatorics and asymptotic analysis; its applications to dynamics, algebra, and other problems will be partly touched upon in this paper, as well as in subsequent publications. The survey contains a lot of new problems from the dynamic theory of graphs and representation theory of groups and algebras. 1.1 A simple example and a difficult question We begin with an elementary example that illustrates the notion of filtration and problems of combinatorial measure theory. Consider the space of all one-sided sequences of zeros and ones: 1 X = fxngn=1 ; xn = 0 _ 1; n = 1; 2;:::; i. e., the infinite product of the two-point space. We will regard X as a dyadic (Cantor-like) compactum in the weak topology, and also as a Borel space. In X we introduce the \tail" equivalence relation and the tail filtration of σ- 0 subalgebras of sets. Namely, two sequences fxkg, fxkg are 0 { n-equivalent if xk+n = xk+n for all k > 0; and { equivalent with respect to the tail equivalence relation if they are n- equivalent for some n. Denote by An, n = 0; 1; 2;::: , the σ-algebra of Borel subsets in X that along with every point x contain all points n-equivalent to x. In other words, An, n = 0; 1; 2;::: , is the σ-subalgebra of Borel sets determined by conditions on the coordinates with indices ≥ n. The decreasing sequence A0 ⊃ A1 ⊃ A2 ⊃ · · · is called the tail Borel filtration of the space X regarded as an infinite direct product, 1 Y X = [0; 1]: n=1 If we endow the space X with an arbitrary Borel measure µ, then the same sequence of σ-subalgebras, which are now understood as σ-subalgebras of classes of sets coinciding mod 0 (with respect to the given measure µ), provides an example of a filtration in a standard measure space. For instance, take µ to be the Bernoulli measure equal to the infinite prod- uct of the measures θ = (1=2; 1=2); the resulting filtration is called the dyadic Bernoulli filtration. By the famous Kolmogorov's zero{one (\all-or-none") law, T the intersection An is the trivial σ-algebra N, which contains only two classes n of sets, of measure either zero or one. 3 For every positive integer n, the space (X; µ) can be represented as a direct product of measure spaces: n Y (X; µ) = f(0; 1); θg × (Xn; µn); 1 1 Y where (Xn; µn) = f(0; 1); θg. n+1 Now we are going to state the main problem. Assume that we have a dyadic compactum X0 with a Borel probability measure µ0 satisfying Kolmogorov's zero{one law, and for every positive integer n there is an isomorphism of measure spaces n 0 0 Y 0 0 (X ; µ ) ' f(0; 1); θg × (Xn; µn); 1 0 0 where (Xn; µn) are some measure spaces. Can we claim that there exists an isomorphism T of measure spaces, T (X0; µ0) = 0 0 (X; µ), for which T (Xn; µ ) = (Xn; µ) for all n? In the language to be described below, this is the question of whether a finitely Bernoulli dyadic ergodic filtration is unique up to isomorphism. It is similar to the following question from the theory of infinite tensor prod- ucts of C∗-algebras and perhaps looks even more natural. Consider a C∗-algebra A and assume that there is a decreasing sequence ∗ An ⊂ A of C -subalgebras of A whose intersection is the space of constants, T An = fConstg, such that for all n n ⊗n A ' M2(C) ⊗ An (here ' means an isomorphism of C∗-algebras). Is it true that there exists an isomorphism 1 Y⊗ A ' M2(C) 1 1 Y ⊗ that sends An to the subalgebra M2(C) ⊂ A for all n? n+1 These problems were raised by the author about fifty years ago. In the origi- nal terms, the first of them looked as follows: is it true that every filtration such that its finite segments are isomorphic to finite segments of a Bernoulli filtration and the intersection of all its σ-algebras is trivial is isomorphic to a Bernoulli filtration? The problem originated from ergodic theory, as we will discuss below. In [60], both questions were answered in the negative; moreover, it turned out that there is a continuum of measure spaces (respectively, C∗-algebras) with dyadic structures that are finitely isomorphic but pairwise nonisomorphic glob- ally. This discovery launched an important field, which, however, still has not 4 gained sufficient attention: the asymptotic theory of filtrations. For this survey, we have selected the most important facts, both known for a long time and new, and, most importantly, new problems in this area. The theory of filtrations has applications to measure theory, random processes, and dynamical systems, as well as to representation theory of groups and algebras. The main problem concerns the complexity of the asymptotic behavior at infinity of monotone se- quences of algebras and a possible classification of such sequences. The second question, about tensor products of C∗-algebras, will be considered in another paper; here we have mentioned it to emphasize the parallelism between prob- lems from very different areas of mathematics. Both these problems belong to asymptotic algebraic analysis. Conceptually new ideas, as compared with the previous research on filtra- tions, appeared in connection with the theory of graded graphs (Bratteli dia- grams); this is one of the central topics of this survey. Apparently, this connec- tion has not been noticed earlier, though the tail filtration is a most important object in the theory of AF-algebras. From the standpoint of the theory of graded graphs, filtrations were considered in the author's recent paper [81]; here, on the contrary, we focus on the measure-theoretic aspect of the problem and use notions and techniques related to graphs. 1.2 Three languages of measure theory Let us return to the measure-theoretic statement of the first problem. If we consider the algebra L1(X; µ) of all classes of measurable bounded functions on the space (X; µ) coinciding mod 0, we obtain a filtration of subalgebras: 1 L (X; µ) ≡ A0 ⊃ A1 ⊃ A2 ⊃ · · · ; 1 where An is the subalgebra in L (X; µ) consisting of all functions that depend on coordinates with indices ≥ n (n = 0; 1; 2;::: ).